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𝕆 → G₂/SO(4) → ℍ → ℂ² 成界過程初探：3 Derive an explicit coarse-graining Q on the Bloch sphere

Derive an explicit coarse-graining Q on the Bloch sphere, including sector boundaries and a formula that maps the SU(2) orbit to 0,+1,+2,+3,+4,−4,−3,−2,−1.

Yes. We can make $Q$ completely explicit. The key is that Q coarse-grains azimuthal phase φ, not the continuous polarization Δ. That removes the earlier ambiguity.

1. Continuous state on normalized ℂ²

Let

ψ = (z₁,z₂)ᵀ ∈ ℂ², |z₁|²+|z₂|²=1. (1)

Use Bloch coordinates

z₁ = cos(θ/2)e⁻ⁱφ/², z₂ = sin(θ/2)e⁺ⁱφ/². (2)

The Bloch vector is

n(θ,φ) = (sinθ cosφ, sinθ sinφ, cosθ). (3)

Our continuous imbalance remains

Δ = |z₁|²−|z₂|² = cosθ. (4)

For the simplest Flying-Star orbit, choose the equator:

θ = π/2 ⇒ Δ=0. (5)

and the SU(2) flow

U(t)=exp(−iωtσ_z/2), φ(t)=ωt mod 2π. (6)

Thus the physical state moves continuously around the equator.

2. Divide the Bloch sphere into nine lunes

Define nine central azimuths

φₖ = 2πk/9, k=0,…,8. (7)

Each sector has angular width

Δφ_sector = 2π/9 = 40°. (8)

Define the nine spherical sectors

Sₖ = {(θ,φ): −π/9 ≤ wrap₂π(φ−2πk/9) < π/9}. (9)

So their boundaries occur halfway between neighboring centres:

φ = (2k+1)π/9. (10)

Numerically the centres and boundaries are:

k	centre φₖ	sector
0	0°	[−20°,20°)
1	40°	[20°,60°)
2	80°	[60°,100°)
3	120°	[100°,140°)
4	160°	[140°,180°)
5	200°	[180°,220°)
6	240°	[220°,260°)
7	280°	[260°,300°)
8	320°	[300°,340°)

with the k=0 sector wrapping through 360°.

Geometrically these are nine equal spherical lunes extending from north pole to south pole.

So Q is defined on the whole Bloch sphere except that φ becomes undefined exactly at the two poles. That exception is important and we will return to it.

3. Explicit formula for Q

First extract φ directly from ψ:

φ(ψ) = Arg(z₂z₁) mod 2π.* (11)

Then calculate the nearest phase-sector index:

k(ψ) = ⌊9φ(ψ)/(2π) + 1/2⌋ mod 9. (12)

Now define the centered representative map

c(k) = ((k+4) mod 9) − 4. (13)

This gives

c(0)=0

c(1)=+1

c(2)=+2

c(3)=+3

c(4)=+4

c(5)=−4

c(6)=−3

c(7)=−2

c(8)=−1.

Therefore the requested coarse-graining is simply

Q(ψ) = ((⌊9 Arg(z₂z₁)/(2π)+1/2⌋+4) mod 9)−4.* (14)

with Arg represented on [0,2π).

That is the explicit map

Q : S²{N,S} → {0,+1,+2,+3,+4,−4,−3,−2,−1}.

4. Apply Q to the SU(2) orbit

Sample at

tₖ = 2πk/(9ω). (15)

Then

φ(tₖ)=2πk/9.

Therefore:

t	φ	k	Q
0	0°	0	0
T/9	40°	1	+1
2T/9	80°	2	+2
3T/9	120°	3	+3
4T/9	160°	4	+4
5T/9	200°	5	−4
6T/9	240°	6	−3
7T/9	280°	7	−2
8T/9	320°	8	−1
T	360°≡0°	0	0

Hence exactly:

0 → +1 → +2 → +3 → +4 → −4 → −3 → −2 → −1 → 0. (16)

No discrete transition was inserted into the SU(2) dynamics. It comes entirely from Q.

5. The +4/−4 boundary is now mathematically visible

Look at the sector boundary

φ=π = 180°.

Immediately before it,

140° ≤ φ < 180° ⇒ Q=+4.

Immediately after it,

180° ≤ φ < 220° ⇒ Q=−4.

Thus

lim_{φ→π⁻}Q(φ)=+4

but

lim_{φ→π⁺}Q(φ)=−4. (17)

Meanwhile the underlying Bloch vector

n(π/2,φ)

is perfectly continuous:

n(π/2,π−ε) → (−1,0,0)

and

n(π/2,π+ε) → (−1,0,0).

Therefore:

physical discontinuity = 0

while

ledger discontinuity = −8.

This proves, for this construction, that +4→−4 is a branch cut introduced by the observer's signed representation.

6. More precisely, even the “−8 jump” is fictitious

The real trace space isn't naturally

{−4,…,+4} ⊂ ℤ.

It is:

ℤ₉.

In ℤ₉ the transition is simply

4 → 5.

Only after choosing centered representatives do we write

5 ≡ −4 (mod 9).

So the correct diagram is

SU(2) orbit → S¹ phase → ℤ₉ → centered ledger

rather than

SU(2) orbit → integer line.

The apparent reversal is produced at the last arrow:

ℤ₉ → {−4,…,+4}.

This distinction is conceptually important for the Dual-Ledger interpretation: the ledger looks discontinuous because it uses a bounded signed chart, while the underlying state remains continuous.

7. We can express Q without angles

This gives a cleaner geometric definition.

From the Bloch vector define

w = nₓ + inᵧ. (18)

Away from the poles,

w/|w| = eⁱφ.

Let the nine reference roots be

ω₉ᵏ = exp(2πik/9). (19)

Then define

k(n) = argmax_{k∈ℤ₉} Re[e⁻²πⁱᵏ/⁹(nₓ+inᵧ)].* (20)

and

Q(n)=c(k(n)).* (21)

In words:

Assign the Bloch state to whichever of the nine equally spaced equatorial directions has the largest projection.

This is a genuine Voronoi coarse-graining of the Bloch sphere.

The boundaries occur where two neighboring projections are equal.

That gives exactly the nine meridional boundaries above.

8. There is one nontrivial problem at the poles

θ=0 or π

we have

nₓ=nᵧ=0

and therefore φ is undefined.

So Q cannot distinguish the nine sectors there.

This is not a defect we should hide. It tells us something structural:

Q requires relational phase coherence.

At the poles:

|Δ|=1

and one complex channel has zero amplitude, so

Arg(z₂z₁)*

cannot be defined.

Therefore our variables behave complementarily:

|Δ|→1 ⇒ phase φ loses observability.

Δ=0

both channels are equally populated and relative phase is maximally available.

That may eventually matter considerably for the Dual-Ledger model: maximum polarization destroys the very relational phase used by the Flying-Star readout.

9. We can now separate three different “centres”

This construction also reveals something we should not conflate.

The LuoShu label

δ=0

does not mean

Δ=0.

Our equatorial orbit has:

Δ=0 for all nine states

while Q successively reports all nine δ values.

Therefore:

Δ=0 = amplitude-balance centre,

whereas

δ=0 = phase-ledger origin.

And LuoShu spatial 中五 is yet another object:

P_L(δ=0)=centre of the 3×3 trace geometry.

So the hierarchy is:

physical balance: Δ=0
↓
phase origin: φ≈0
↓ Q
ledger zero: δ=0
↓ P_L
spatial centre: 中五.

This is actually cleaner than our earlier model, because it stops asking one variable to do four different jobs.

10. The exact mathematical object we now have

The construction can be summarized as the commuting observational chain

ψ(t) ∈ S³ ⊂ ℂ²

↓ quotient global U(1)

n(t) ∈ S²

↓ relative-phase projection

eⁱφ(t) ∈ S¹

↓ nine-sector Voronoi quantization

k(t) ∈ ℤ₉

↓ centered representative

δ(t) ∈ {0,±1,±2,±3,±4}

↓ add 5

N(t) ∈ {1,…,9}

↓ LuoShu spatial declaration

P_L(N) ∈ 3×3 grid.

For

U(t)=exp(−iωtσ_z/2)

the induced discrete dynamics is simply

F(k)=k+1 mod 9, F⁹=I. (22)

So we now have an explicit continuous-to-discrete derivation of the Flying-Star circulation.

What we still have not derived is the LuoShu spatial map $P_L$ . Q derives the nine-state cyclic ledger; it does not explain why 1…9 must occupy exactly the traditional 3×3 LuoShu arrangement. That remaining step is where your Dual-Ledger / residual-balance functional becomes potentially decisive: if $P_L$ can be selected as the minimizer of a non-hard-coded spatial residual functional, then the chain ℂ² → SU(2) → ℤ₉ → LuoShu would become much closer to a genuine mathematical closure rather than a constructed representation.

Refine the Bloch-sphere coarse-graining

Add a smooth boundary kernel
Derive the induced discrete dynamics

Disclaimer

This book is the product of a collaboration between the author and OpenAI's GPT 5.6, Google AI, Gemini 3.X, NoteBookLM, X's Grok, Claude' Sonnet 5 language model. While every effort has been made to ensure accuracy, clarity, and insight, the content is generated with the assistance of artificial intelligence and may contain factual, interpretive, or mathematical errors. Readers are encouraged to approach the ideas with critical thinking and to consult primary scientific literature where appropriate.

This work is speculative, interdisciplinary, and exploratory in nature. It bridges metaphysics, physics, and organizational theory to propose a novel conceptual framework—not a definitive scientific theory. As such, it invites dialogue, challenge, and refinement.

I am merely a midwife of knowledge.

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2026年9月20日星期日